You can sort elements using two stacks by systematically pushing and popping elements between them in a defined manner.

To sort the elements using two stacks, you start with all the elements in the first stack, with the objective of transferring all these elements to the second stack in sorted order. The process involves repeatedly popping the top element from the first stack and comparing it with the top element of the second stack.

Here are the steps involved:

1. Pop the Top Element: Begin by popping the top element from the first stack.
2. Comparison with Second Stack:
   • If the second stack is empty, or if the popped element is greater than the top element of the second stack, push the popped element onto the second stack.
   • If the popped element is less than the top element of the second stack, pop elements from the second stack and push them back onto the first stack until you find the appropriate position for the popped element in the second stack.
3. Repeat: Continue this process until all elements from the first stack have been successfully transferred to the second stack.

At the conclusion of this process, the elements in the second stack will be sorted, with the smallest element positioned at the top.

This sorting technique is referred to as stack sort. It is straightforward and efficient for sorting small datasets; however, it becomes less effective for larger datasets due to the extensive number of push and pop operations required, which can be time-consuming.

Time and Space Complexity

In terms of time complexity, stack sort has a worst-case and average time complexity of $O(n^2)$, where $n$ represents the number of elements. This quadratic complexity arises because, in the worst-case scenario, for each element we may need to pop all elements from the second stack and push them back onto the first stack. The space complexity is $O(n)$, as we utilize two stacks to store the elements.

Conclusion

In summary, sorting elements using two stacks is a simple and effective approach for handling small datasets. However, due to its time complexity, it may not be the most efficient method for larger datasets.